Sur une condition suffisante pour l’existence de mesures p-adiques admissibles
Journal de théorie des nombres de Bordeaux, Tome 15 (2003) no. 3, pp. 805-829.

On donne une nouvelle condition suffisante pour l’existence des mesures p-adiques admissibles μ obtenues à partir de suites de distributions Φj(j≥0) à valeurs dans les espaces de formes modulaires. On utilise la projection caractéristique sur le sous-espace primaire associé à une valeur propre non nulle α de l’opérateur U d’Atkin. Notre condition est exprimée en termes des congruences entre les coefficients de Fourier des formes modulaires Φj. On montre comment vérifier ces congruences, et on traite plusieurs applications. On obtient donc une explication conceptuelle des formules de Yu.Manin pour les distributions attachées à la fonction Lf(s,χ)=∑n≥1χ(n)ann-s d’une forme parabolique primitive f=∑n≥1anqn∈Sk(Γ0(N),ψ) de poids k≥2.

We give a new sufficient condition for the existence of admissible p-adic measures μ obtained from sequences of distributions Φj(j≥0) with values in spaces of modular forms. We use the characteristic projection on the primary subspace associated to a non zero eigenvalue α of the Atkin operator U. Our condition is expressed in terms of congruences between the Fourier coefficients of the modular forms Φj. We show how to verify these congruences and we give several applications. So we get a conceptual explanation for the Yu.Manin’s formulas for the distributions attached to the L-function, Lf(s,χ)=∑n≥1χ(n)ann-s, of a primitive cuspform f=∑n≥1anqn∈Sk(Γ0(N),ψ) of weight k≥2.

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     author = {Panchishkin, Alexei},
     title = {Sur une condition suffisante pour l{\textquoteright}existence de mesures $p$-adiques admissibles},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {805--829},
     publisher = {Universit\'e Bordeaux I},
     volume = {15},
     number = {3},
     year = {2003},
     mrnumber = {2142237},
     zbl = {1078.11038},
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     url = {https://www.numdam.org/item/JTNB_2003__15_3_805_0/}
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Panchishkin, Alexei. Sur une condition suffisante pour l’existence de mesures $p$-adiques admissibles. Journal de théorie des nombres de Bordeaux, Tome 15 (2003) no. 3, pp. 805-829. https://www.numdam.org/item/JTNB_2003__15_3_805_0/

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